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Mirrors > Home > MPE Home > Th. List > xlt0neg2 | Structured version Visualization version GIF version |
Description: Extended real version of lt0neg2 11661. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xlt0neg2 | ⊢ (𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 11201 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | xltneg 13135 | . . 3 ⊢ ((0 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (0 < 𝐴 ↔ -𝑒𝐴 < -𝑒0)) | |
3 | 1, 2 | mpan 688 | . 2 ⊢ (𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < -𝑒0)) |
4 | xneg0 13130 | . . 3 ⊢ -𝑒0 = 0 | |
5 | 4 | breq2i 5113 | . 2 ⊢ (-𝑒𝐴 < -𝑒0 ↔ -𝑒𝐴 < 0) |
6 | 3, 5 | bitrdi 286 | 1 ⊢ (𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 class class class wbr 5105 0cc0 11050 ℝ*cxr 11187 < clt 11188 -𝑒cxne 13029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-xneg 13032 |
This theorem is referenced by: xmulneg1 13187 xrhmeo 24307 |
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